This tag is for questions about the foundations of mathematics, and the formalization of mathematical concepts in foundational theories (e.g. set theory, category theory, and type theory).

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Bachmann's construction of the real numbers

On page 44 of this book an approach to constructing the real numbers as equivalence classes of nested rational intervals is outlined and attributed to Bachmann. The outline in the book is very ...
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1answer
71 views

In Whitehead & Russell's PM, what makes anything true or false?

It seems that truth and falsehood are fundamental to meanings and types. A proposition is defined as anything that is true or that is false. PM defines truth as "consisting in the fact that there is ...
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66 views

Do all mathematical fields require an algebra?

My understanding is that "algebra" refers to a specific field in mathematics. Here is Wikipedia's introduction: Algebra is one of the broad parts of mathematics, together with number theory, ...
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1answer
99 views

Why don't we use Presburger's arithmetic instead of Peano's arithmetic?

I was reading about quantifier elimination and discovered the Presburger Arithmetic, the article mentions two points about it: It is decidable, complete and consistent. It omits multiplication ...
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1answer
62 views

Foundations of math and primitive terms leading to Russells paradox

If the statement "Is an element of" is a primitive term (is not defined/you cant determine its truth) then how do you determine the truth of statements such as "If $x$ is in $S$, then $x$ has property ...
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3answers
194 views

Intuitionistic Logic and Classical Logic on the proof of (A or B)

In intuitionist logic, a proof of (A or B) means a proof of A, or a proof of B, whereas in Classical logic, a proof of (A or B) may be done withouth either proving A or proving B. I'm trying to ...
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1answer
64 views

What does if-then has to do with not being true?

I'm reading Chihara's: Constructibility and Mathematical Existence. It says: An even more radical view rejects the assumption that mathematics is true—at least in the straightforward way that ...
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1answer
42 views

A question on Mortiz Pasch works in foundation of mathematics

I am reading "Introduction to the Foundation of Mathematics" ,by R L.Wilder(2nd Ed.), where Mortiz Pasch's works are described in paragraph 1.5. There is a quotation in that paragraph- " For if,on ...
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1answer
79 views

Modern algebra and set theory: ZFC vs. NBG

This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little: Is it not more natural consider NBG set theory as the foundation ...
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1answer
180 views

True or false? If $\eta$ is an explicitly defined incomputable number, then no formal system can pin down the value $\eta$ to arbitrary precision.

Let $\eta$ denote an explicitly defined incomputable real number (the bounty text is faulty, and does not mention incomputability of $\eta$). Then I think that no (recursively ennumerable) formal ...
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6answers
1k views

What's behind the Banach-Tarski paradox? [closed]

The discovery of the Banach-Tarski paradox was of course a great thing in mathematics but raises the issue of the relation between mathematics and reality. Empirically there are good reasons for faith ...
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0answers
50 views

Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. ...
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2answers
256 views

What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
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1answer
46 views

What are the leading alternative foundations for mathematics?

I know that all of mathematics can be recast in terms of set theory. There are multiple choices for this set theory (some form of ZFC, NBG, NF, etc.), and so multiple possible set theoretic ...
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1answer
43 views

How can the two basic binary operations (addition and subtraction) be defined in set-theoretical terms?

I recently stumbled upon this interesting definition of mathematics: Math is the study of things that can be described as sets I am aware that the integers and the real numbers can be defined in ...
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25 views

Relations- Find the inverse of a relation

Find an inverse for 41 modulo 660. I don't really understand what is the question asking. But I know, what is the meaning of inverse relation.eg (x,y) ϵR ,then (y,x) ϵ R^-1. Anyone can explain to ...
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1answer
119 views

How to define mathematical objects from scratch?

Are there any "Zermelo-type theories" for some other type of mathematical objects except for sets? I.e., axiomatic systems for mathematical objects formalized for first-order logic using a language ...
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0answers
59 views

What is the standard foundation for category theory?

Reference : http://arxiv.org/abs/0810.1279 For 2 days, I have searched what possible foundations for category theory are there and I think the reference link explains possible foundations nicely. The ...
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2answers
459 views

How are the elementary arithmetics defined?

In the book Principles of Mathematical Analysis by Rudin, I read that "a < b" is defined this way: if b - a is positive, then a < b or b > a. Then some questions arose to me: we know that ...
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1answer
55 views

Kolmogoroff's Axioms of Probability and Completness

In Kolmogoroff Classic Foundations of the Theory of Probability, right at the beginning he gives the (now well-known axioms) Let $E$ be a collection of elements $\xi,\eta,\zeta,\ldots$ which we ...
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88 views

Intuition for the choice of background (set) theory

Problem From the formalist point of view, any mathematical statement should ultimately be an assertion about the derivability of a certain formula in a certain formal system, call it the background ...
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1answer
69 views

Most general mathematical framework

One can think of the same mathematical object in many different ways. For example take $\mathbb{R}$. One can think of this as (assume necessary hypotheses and so on) As a group. As a one ...
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1answer
79 views

The fundamental axioms of mathematics

Having known about what axioms are, I want to know whether there are some "fundamental axioms of mathematics" on which every branch of mathematics depends. If yes, what are they ? Or Do we have ...
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1answer
72 views

What is the difference between logical and iterative set

Saphiro in his "foundations without foundationalism: a case for second-order logic" defends second-order logic by claiming that talking about subsets of domain is not problematic in case of SOL. He ...
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1answer
77 views

In Whitehead & Russell's PM, if $P$ is an infinite well-ordered series, can $P$ have a last term?

If I'm not mistaken, $B‘\overset{\smile}{P}$ is the last term of $P$. If it does not exist, there is no need to put ~$(B‘\overset{\smile}{P}) \in C‘∇‘P $ in the hypothesis. Chances are I missed ...
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1answer
156 views

Category theory? Logic? Anyone experienced this like me? [closed]

Mathematics is not logic, but if one uses Zorn's lemma and stuff he should accept logical impact on mathematics. I'm the one who cares a lot about logics. It seems like Category theory is inevitable ...
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3answers
135 views

In Whitehead & Russell's PM, does every Series contain a $P_1$ (immedeately precedes)?

✳204.7 $\vdash: P \in Ser .\supset. P_1 \in 1 \rightarrow 1$ Which says if $P$ is a series, then $P_1$ is one-one. ✳201.63 $\vdash: P \in trans \cap Rl‘J .\supset. P_1 = P \overset{.}{-}P^2$ ...
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0answers
98 views

An adequate difference between $\forall x\in A:P(x)$ and $(\forall x)(x\in A\rightarrow P(x))$?

Ever since I was a young student I have felt doubts about the traditional $(\forall x)$-expression: starting a statement with such an irrational lack of focus doesn't seems reasonable! I mean, all $x$ ...
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1answer
88 views

Question about epsilon-delta definition of limits.

In Chapter 1: Functions and limits, 1.7 The Precise Definition of a Limit, Let $f$ be a function ... the limit of $f(x)$ as $x$ approaches $a$ is $L$, and we write $$\lim_{x\to a }f(x)=L$$ if ...
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0answers
74 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
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2answers
44 views

What are the ramifications of introducing a universal set this way?

What are the ramifications of introducing a universal set using this axiom? $$\exists x : \forall y (y\neq x \rightarrow y\in x)$$
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3answers
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What in Mathematics cannot be described within set theory? [duplicate]

I have begun reading Patrick Suppes' book Axiomatic Set Theory. The first sentence in chapter 1 reads: "Among the many branches of modern mathematics set theory occupies a unique place: with a few ...
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3answers
140 views

quantification domain of set theory formulas

Let ZFC set theory, what is the domain of quantification of a formula like $\forall x\phi(x)$? If the domain is the whole Von Neumann Hierarchy $V$ why it is not a problem that it doesn't form a set?
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1answer
62 views

In Whitehead&Russell's PM's ✳210, how can the product of $\lambda$ be not a member of $\lambda$?

Take ✳210.23 for example: Assuming $\kappa$ is a classes of classes such that, of any two, one is contained in the other, i.e. $\alpha, \beta \in \kappa .\supset_{\alpha, \beta} : \alpha \subset ...
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1answer
34 views

In Whitehead&Russell's PM, What is $\max_p$'s converse domain?

Here is the definition of upper limit. If I'm not mistaken, $\max_P$'s converse domain is the universal set $V$. The definition appears to be limiting the converse domain of $\operatorname{seq}_P$ ...
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5answers
1k views

What is the meaning of set-theoretic notation {}=0 and {{}}=1?

I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
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1answer
388 views

What axioms does ZF have, exactly?

While trying to find the list of axioms of ZF on the Web and in literature I noticed that the lists I had found varied quite a bit. Some included the axiom of empty set, while others didn't. That is ...
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In Whitehead and Russell's PM, are homogenous relations the only ones that have relation numbers?

Given the definition of ordinal similarity: ✳151.01 $P \overline{smor} Q = \hat{S}\{ S\in 1\rightarrow 1. C‘Q=ConverseD‘S. P=S^;Q\}$ Df. $Q$ has to be homogeneous, otherwise $C‘Q$ is meaningless. ...
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1answer
81 views

What are sets and classes in maths, and how are they related to $O()$ and $o()$ notation?

Are there many definitions of sets and classes in mathematics, as given in Formal definion of the notations used in measuring time complexity? And in particular, why the notation given in Fedja's ...
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2answers
90 views

Is it meaningless to say $M\prec N$ for two proper class models?

Kunen in page 88 of his "Set Theory" book says: ... For a specific given $\varphi$, the notion $M\prec_{\varphi}N$ (i.e. $\forall \overline{a}\in M~~~M\models \varphi ...
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4answers
378 views

Are there counter intuitive interpretations of ZF or ZFC?

Are there interpretations of ZF or ZFC that are non standard in the sense that $\epsilon$ is interpreted in a counter intuitive way that intuitively has nothing to do with "belongs to" or "is part of" ...
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3answers
193 views

Models of set theory

How can one talk of a semantics or model of set theory (lets say ZF or ZFC) when the definition of a structure (and potential model) needs a carrier set in the first place (by its definition)?
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2answers
111 views

How do you multiply infinite quantities?

Out of curiosity I was watching this video from njwildberger on youtube: https://www.youtube.com/watch?v=4DNlEq0ZrTo Where he says that you can't define associativity between irrational numbers ...
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1answer
113 views

How much maths can we do in NF(U)?

I have recently become interested in non-standard set theories, particularly in NF and NFU and have been reading some things here and there. Of course, I don't know much about it and I'm still trying ...
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0answers
49 views

Propositions as sets of witnesses

Under the propositions-as-types paradigms, a proposition is identified with the type of all its proofs. From a more classical perspective (and assuming the full-blown axiom of choice), it sometimes ...
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4answers
315 views

A question regarding ❋166.44 in Whitehead & Russell's Principia Mathematica

In the first step of Dem, I wonder how $\Sigma ‘\times P^{;}Q$ is transformed into $\Sigma‘ \Sigma^;(P \overset{\downarrow}{.,})\dagger^; Q$. Thanks,
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1answer
181 views

Is there a reasonably strong foundation for mathematics that can prove its on consistency?

Ever since I have read about both Gödel's incompleteness theorem(s?), which I believe roughly means: "A system at least as strong as Peano arithmetic cannot prove its own consistency." and learned ...
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2answers
133 views

Well Formed Expression (Polish Notation)

In Kunen's book Foundation of Mathematics the definition of a well formed expression (wfe) of a lexicon for Polish notation $\langle W, \alpha \rangle$ ($W$ is a set and $\alpha:W\to\omega$ is a ...
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53 views

Codifying ways to think and write about imprecise ideas?

This question will not be about affine spaces; rather those are mentioned here as one of many examples. A vector space has an underlying set and a field of scalars and an operation of linear ...
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2answers
171 views

Well-ordering principle and negative integers

The Wikipedia article on the Well Ordering Principle defines it [1] as: "The well-ordering principle states that every non-empty set of positive integers contains a least element." And it defines ...